Prove the following statement If S is a nonempty subset of R

Prove the following statement. If S is a nonempty subset of R_n, then (S) = span(S).

Solution

If S Rⁿ , then the orthogonal complement of S, denoted S, is the set of all vectors x Rⁿ that are orthogonal to S i.e., S is the largest subset of Rⁿ which is orthogonal to S. Further,   S = (S) . Hence Span(S) = S(S) . (For any subspace V of Rⁿ, we have (V) = V